Advertisement

Hopf surfaces in locally conformally Kähler manifolds with potential

  • Liviu OrneaEmail author
  • Misha Verbitsky
Original Paper
  • 3 Downloads

Abstract

An LCK manifold with potential is a quotient M of a Kähler manifold X equipped with a positive plurisubharmonic function f, such that the monodromy group acts on X by holomorphic homotheties and maps f to a function proportional to f. It is known that a compact M admits an LCK potential if and only if it can be holomorphically embedded to a Hopf manifold. We prove that any non-Vaisman, compact LCK manifold with potential contains a complex surface (possibly singular) with normalization biholomorphic to a Hopf surface H. Moreover, H can be chosen non-diagonal, hence, also not admitting a Vaisman structure.

Keywords

Locally conformally Kähler Potential Hopf manifold Vaisman manifold 

2000 Mathematics Subject Classification

53C55 

Notes

Acknowledgements

L.O. thanks the Laboratory for Algebraic Geometry at the Higher School of Economics in Moscow for hospitality and excellent research environment during February and April 2014, and April 2015. Both authors are indebted to Paul Gauduchon, Andrei Moroianu, and Victor Vuletescu for extremely useful disussions.

References

  1. 1.
    Apostolov, V., Dloussky, G.: Locally conformally symplectic structures on compact non-Kähler complex surfaces. IMRN 9, 2717–2747 (2016). arXiv:1501.02687 CrossRefGoogle Scholar
  2. 2.
    Atyiah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley, Reading (1969)Google Scholar
  3. 3.
    Belgun, F.A.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317(1), 1–40 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brunella, M.: A characterization of Inoue surfaces. Comment Math. Helv. 88(4), 859–874 (2013). arXiv:1011.2035 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chiose, I., Toma, M.: On compact complex surfaces of Kähler rank one. Am. J. Math. 135(3), 851–860 (2013). arXiv:1010.2591 CrossRefGoogle Scholar
  6. 6.
    Dragomir, S., Ornea, L.: Locally Conformally Kähler Geometry. Progress in Mathematics, vol. 155. Birkhäuser, Basel (1998)CrossRefGoogle Scholar
  7. 7.
    Gauduchon, P.: La \(1\)-forme de torsion d’une variété Hermitienne compacte. Math. Ann. 267, 495–518 (1984)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Harvey, R., Lawson Jr., H.B.: An intrinsic characterization of Kähler manifolds. Invent. Math. 74(2), 169–198 (1983)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Humphreys, J.E.: Linear Algebraic Groups, GTM 21, 4th edn. Springer, New York (1998)Google Scholar
  10. 10.
    Istrati, N., Otiman, A.: De Rham and twisted cohomology of Oeljeklaus–Toma manifolds. Ann. Inst. Fourier 69(5), 2037–2066 (2019). arXiv:1711.07847 CrossRefGoogle Scholar
  11. 11.
    Kasuya, H.: Vaisman metrics on solvmanifolds and Oeljeklaus–Toma manifolds. Bull. Lond. Math. Soc 45(1), 15–26 (2013). arXiv:1204.1878 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kato, M.: Some remarks on subvarieties of Hopf manifolds. Tokyo J. Math. 2(1), 47–61 (1979)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kokarev, G.: On pseudo-harmonic maps in conformal geometry. Proc. Lond. Math. Soc. 99, 168–94 (2009). arXiv:0705.3821 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kollár, J.: Exercises in the Birational Geometry of Algebraic Varieties. Analytic and Algebraic Geometry, IAS/Park City Mathematics Series, vol. 17, pp. 495–524. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  15. 15.
    Moroianu, A., Moroianu, S.: On pluricanonical locally conformally Kähler manifolds. Int. Math. Res. Not. 2017(14), 4398–4405 (2017).  https://doi.org/10.1093/imrn/rnw151. arXiv:1512.04318
  16. 16.
    Nakamura, I.: On surfaces of class VII\({}_0\) with curves. Invent. Math. 78, 393–443 (1984)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Oeljeklaus, K., Toma, M.: Non-Kähler compact complex manifolds associated to number fields. Ann. Inst. Fourier 55, 1291–1300 (2005)CrossRefGoogle Scholar
  18. 18.
    Ornea, L., Verbitsky, M.: Locally conformal Kähler manifolds with potential. Math. Ann. 348, 25–33 (2010). arXiv:math/0407231 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Ornea, L., Verbitsky, M.: Topology of Locally Conformally Kähler Manifolds with Potential, IMRN, vol. 2010, pp. 717–726. arXiv:0904.3362
  20. 20.
    Ornea, L., Verbitsky, M.: Locally conformally Kahler metrics obtained from pseudoconvex shells. Proc. Am. Math. Soc. 144, 325–335 (2016). arXiv:1210.2080 CrossRefGoogle Scholar
  21. 21.
    Ornea, L., Verbitsky, M.: LCK rank of locally conformally Kähler manifolds with potential. J. Geom. Phys. 107, 92–98 (2016). arXiv:1601.07413 MathSciNetCrossRefGoogle Scholar
  22. 22.
    Otiman, A.: Morse-Novikov cohomology of locally conformally Kähler surfaces. Mathematische Zeitschrift 289(1–2), 605–628 (2018). arXiv:1609.07675 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Vaisman, I.: Generalized Hopf manifolds. Geom. Dedicata 13(3), 231–255 (1982)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Verbitsky, M.: Theorems on the vanishing of cohomology for locally conformally hyper-Kähler manifolds. Proc. Steklov Inst. Math. 246(3), 54–78 (2004). arXiv:math/0302219 Google Scholar
  25. 25.
    Verbitskaya, S.M.: Curves on the Oeljeklaus–Toma manifolds. Funct. Anal. Appl. 48(3), 223–226 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BucharestBucharestRomania
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  3. 3.Instituto Nacional de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  4. 4.Laboratory of Algebraic Geometry, Department of MathematicsHSE UniversityMoscowRussia

Personalised recommendations